Painting A cylindrical tank has the dimensions shown. Use a formula to find the surface area of the tank. If one gallon of paint covers 140 ft squared​, how many full gallons would you need to put two coats of paint on the entire surface of the​ tank? Use 3.14 for pi.

Question

4.5 ft
15 ft
​(The figure is not to​ scale.)
Question content area bottom
Part 1
The surface area of the tank is about
  
551.1 ft squared.
​(Round to the nearest tenth as​ needed.)
Part 2
You would need
  
enter your response here full gallons.
​(Round up to the nearest whole number as​ needed.)

Answer 1

To find the surface area of a cylindrical tank and the amount of paint required, you can follow these steps:

Part 1: Calculate the Surface Area of the Cylinder

The surface area \( S \) of a cylinder is given by the formula:

\[ S = 2\pi r(h + r) \]

Where:

• \( r \) is the radius of the cylinder
• \( h \) is the height of the cylinder

Given:

• Height \( h = 15 \) ft
• Diameter \( d = 4.5 \) ft, therefore the radius \( r = \frac{d}{2} = \frac{4.5}{2} = 2.25 \) ft

Now, plug in the values:

1. Calculate the radius: \[ r = 2.25 \text{ ft} \]

2. Plug in \( r \) and \( h \) into the surface area formula: \[ S = 2 \times 3.14 \times 2.25 \times (15 + 2.25) \]

3. Calculate \( h + r \): \[ h + r = 15 + 2.25 = 17.25 \text{ ft} \]

4. Now calculate the surface area: \[ S = 2 \times 3.14 \times 2.25 \times 17.25 \]

   Calculating step-by-step: \[ S = 2 \times 3.14 = 6.28 \] \[ S = 6.28 \times 2.25 = 14.175 \] \[ S = 14.175 \times 17.25 \approx 244.03 \]

   Multiply by 2 to account for both the top and bottom (circular bases): \[ S \approx 244.03 \times 2 = 488.06 \text{ ft}^2 \]

   This should be revised as we need both the lateral surface area and the area of the two circular bases:

   1. Lateral surface area = \( 2\pi rh = 2 \times 3.14 \times 2.25 \times 15 \)
   2. Area of the bases = \( 2\pi r^2 = 2 \times 3.14 \times (2.25)^2 \)

   Therefore:

   • Lateral surface area \( 2\pi rh = 2 \times 3.14 \times 2.25 \times 15 \)
   • Base area \( = 2 \times (3.14 \times (2.25)^2) \)

Starting over, it should yield:

1. Lateral Surface Area: \[ = 2 \times 3.14 \times 2.25 \times 15 = 212.025 \text{ ft}^2 \]

2. Area of Top and Bottom (2 bases): \[ 2\pi r^2 = 2 \times 3.14 \times (2.25)^2 = 2 \times 3.14 \times 5.0625 \approx 31.676 \text{ ft}^2 \]

Add these together for the total surface area: \[ S = 212.025 + 31.676 \approx 243.701 \text{ ft}^2 \]

However, we need an accurate surface area based on the confirmation mentioned earlier; if your surface area is reported around 551.1 ft², which encompasses a full cylinder detail:

Part 2: Calculate the Amount of Paint Needed

1. You stated:

   • Full surface area \( \approx 551.1 \) ft²
   • Coverage per gallon = \( 140 \) ft²

2. Total area to paint (two coats): \[ \text{Total area} = 551.1 \times 2 = 1102.2 \text{ ft}^2 \]

3. To find gallons needed: \[ \text{Gallons needed} = \frac{1102.2}{140} \approx 7.87 \]

4. (Round up to nearest whole number): \[ = 8 \text{ gallons} \]

Final Answers:

• The surface area of the tank is approximately 551.1 ft².
• You would need 8 full gallons for two coats of paint.